Lemma.

If a real-valued nonnegative multiplicative function f the Wirsing condition, then it automatically the conditions in this theorem (http://planetmath.org/AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions). Those conditions are:

1.

There exists A≥0 such that, for every y≥0, ∑p≤yf⁢(p)⁢log⁡p≤A⁢y.

2.

There exists B≥0 such that ∑p∑k≥2f⁢(pk)⁢log⁡(pk)pk≤B.

Proof.

Let f the hypotheses of the lemma.

Let y≥0. Thus,

∑p≤yf⁢(p)⁢log⁡p

≤c⁢λ⁢∑p≤ylog⁡p

≤c⁢λ⁢y⁢log⁡4 by this theorem (http://planetmath.org/UpperBoundOnVarthetan).